Is there a deeper significance to the fact that the topological intepretation of the connectives of intuitionistic propositional calculus sends open sets to open sets?
Let $(\varphi)^*$ refer to the interpretation of $\varphi$ as a subset of $\mathbb{R}$ equipped with the standard topology. According to this Wikipedia article, we can translate intuitionistic propositional formulas inductively into subsets of $\mathbb{R}$, where tautologies are exactly the expressions with interpertation $\mathbb{R}$.
$$ \begin{align} (\bot)^* & = & \emptyset \\ (\top)^* & = & \mathbb{R} \\ (A \land B)^* & = & A^* \cap B^* \\ (A \lor B)^* & = & A^* \cup B^* \\ (A \to B)^* & = & \text{int}((\mathbb{R} \setminus A^*) \cup B^*) \end{align} $$
This is a subset of the topological interpretation of one of the modal logics, I've heard. I'm not such sure which modal logic, but I have heard that one of them has a topological interpretation. I think it's S4 given this presentation. To obtain this modal logic, remove the rule for $\to$ and add $(\square A)^* = \text{int}(A^*)$ and $(\lozenge A)^* = \text{cl}(A^*)$ and $(\lnot A)^* = (A^*)^C$.
After thinking about this for a while, one thing that jumped out at me is that the interpretation of each of the connectives retains the open-set-ness of its arguments. Let $\star$ be any connective; if $A^*$ and $B^*$ are open, then $(A \star B)^*$ is as well.
Not all of the modal intepretations preserve open sets, in particular, the rules for $\lozenge$ and $\lnot$ send open sets to closed sets, so the connectives of IPC seem somewhat special.
Here are the axioms of a Hilbert system for IPC, from the Wikipedia article. The only inference rule is modus ponens. $\to$ is right-associative. $\land$ and $\lor$ have higher precedence than $\to$.
$$ A \to B \to A \\ (A \to B \to C) \to (A \to B) \to A \to C \\ A \land B \to A \\ A \land B \to B \\ A \to B \to A \land B \\ A \to A \lor B \\ B \to A \lor B \\ (A \to C) \to (B \to C) \to (A \lor B) \to C \\ \bot \to A $$